Two Solutions for the Work-Kinetic Energy Theorem and the Principle of Conservation of Energy

This paper presents a complete analytical development of one of the most frequently used problems, in describing the way in which the energy method is applied on the solution of problems on basic physics and engineering courses. The majority of the classical references do not explain the meaning of finding two solutions when using the principle of conservation of mechanical energy. The classical references just consider the existence of one real solution. The problem corresponds to the displacement of a body on an inclined plane with dry friction, the body moves by the action of gravity and by an initial condition of velocity that is given to it. At the lower end of the inclined plane there is a previously compressed spring, that produces an initial force when the body comes into contact with the spring, it is required to determine the maximum displacement that the body will reach, once it comes into contact with the spring. The use of the Newton equations completes and contributes to explain the meaning of the second solution of the problem, to which corresponds with the real behavior of the system.